A new systematic analysis technique was developed using the Effective Field Theory (EFT) approach for direct dark matter detection and folding in energy-dependent information, when possible, about the detected events, experimental efficiencies, and backgrounds. The Bayesian inference tool, MultiNest, efficiently calculates highly-dimensional likelihoods over WIMP mass and multiple EFT coupling coefficients. The resulting likelihoods can then be used to set limits on these parameters and choose models (EFT operators) that best fit the direct detection data. Expanding the parameter space beyond the standard spin-independent isoscalar cross-section and WIMP mass reduces tensions between previously published experiments. Combining these experiments into a single likelihood leads to stronger limits than when each experiment is considered on its own. Simulations using two non-standard operators (operators 3 and 8) are used to test the proposed analysis technique in up to five dimensions and demonstrate the importance of using multiple likelihood projections when determining the limits on WIMP mass and EFT coupling coefficients.